多项式板子

多项式理论和模板

理论部分

dft(fft/ntt)

对于多项式F(x)F(x)带入单位复根WnkW_n^k可以得到点值，然而求每一个点值消耗的时间是O(d(F))O(d(F))的，通过考虑分治法，把

F(x)=a0+a1⋅x+...an⋅xnF(x)=a_0+a_1\cdot x+...a_n\cdot x^n

进行奇偶分类，得到

G(x)=a0+a2⋅x+...an⋅x(n−1)/2,H(x)=a1+a3⋅x+...+an⋅x(n−1)/2G(x)=a_0+a_2 \cdot x+...a_{n} \cdot x^{(n-1)/2},H(x)=a_1+a_3 \cdot x+...+a_n \cdot x^{(n-1)/2}

那么可以得到

F(x)=G(x2)+x∗H(x2)F(x)=G(x^2)+x*H(x^2)

注意到偶数次单位复根的性质：

Wnk=−Wnk+n/2W_n^k=-W_n^{k+n/2}

因为G和H都为x2x^2的多项式，所以这两个复根的值相同，那么只要计算出来 G(wn/2k)G(w_{n/2}^{k}) 和 H(wn/2k)H(w_{n/2}^{k}) 之后就可以算出来F(wnk)F(w_n^k) 和 F(wnk+n/2)F(w_n^{k+n/2})的值，那么就可以进行递归计算了。ps.在mod n 域上的多项式其单位根可以是膜数的单位根，其与复数单位根性质类似。

位逆序置换

要对dft进行分治法加速，那就要让多项式项数变成2的幂次。

MTT

方法1:进行三次不同模数的NTT，然后就可以算出在其他模数下的NTT结果。

方法2:进行拆系数fft，可以提高fft的精度，以便其在模意义下有足够精度。

多项式求逆

对于多项式 F[x]∗G[x]=1F[x]*G[x]=1 ,假设 F[x]∗H[x]=1(mod  xn)F[x]*H[x]=1(\mod x^n) 那么就有

F[X](G[X]−H[X])=0(mod  xn)F[X](G[X]-H[X])=0( \mod x^n)

考虑两边平方，就有

(G[X]−H[X])2=0(mod  x2n)(G[X]-H[X])^2=0(\mod x^{2n})

左右同时乘以F[X]F[X]得到，

F[X]∗G[X]∗G[X]−2∗F[X]∗G[X]∗H[X]+F[X]∗H[X]∗H[X]=1(mod  x2n)F[X]*G[X]*G[X]-2*F[X]*G[X]*H[X]+F[X]*H[X]*H[X]=1( \mod x^{2n})

因为G为F逆元，那么G[X]−2H[X]+F[X]∗H[X]∗H[X]=0(mod  x2n)G[X]-2H[X]+F[X]*H[X]*H[X]=0(\mod x^{2n})因为H[0]=F[0].inv()H[0]=F[0].inv()那么通过n从1开始倍增，就可以求出G。
也可以通过牛顿迭代得到。

多项式除法

给定F(x)=G(x)∗Q(x)+R(x)F(x)=G(x)*Q(x)+R(x)

和F(x)F(x),G(x)G(x)求Q(x),R(x)Q(x),R(x)

其中F(x)F(x)为n阶多项式G(x)G(x)为m阶多项式R(x)R(x )为小于m阶的多项式。

那么对于F(1x)=G(1x)∗Q(1x)+R(1x)F(\frac {1}{x})=G(\frac{1}{x})*Q(\frac{1}{x})+R(\frac {1}{x})xnF(1x)=xmG(1x)∗xn−mQ(1x)+xn−m+1⋅(xm−1R(1x))x^nF(\frac{1}{x})=x^{m}G(\frac{1}{x})*x^{n-m}Q(\frac{1}{x})+x^{n-m+1}\cdot (x^{m-1}R(\frac{1}{x}))

因为xnF(1x)=Fr(x)x^nF(\frac{1}{x})=F_r(x)(Fr(x)F_r(x)代表翻转之后的F(x)F(x))

所以上面的式子可以化成Fr(x)=Gr(x)∗Qr(x)+xn−m+1Rr(x)F_r(x)=G_r(x)*Q_r(x)+x^{n-m+1}R_r(x)

对其同时对于xn−m+1x^{n-m+1}取模，得到Fr(x)=Gr(x)∗Qr(x)(mod  xn−m+1)F_r(x)=G_r(x)*Q_r(x)(\mod x^{n-m+1})

因为G(x)G(x)已知，可以通过多项式求逆，得到Q(x)Q(x),然后通过原始公式得到R(x)R(x)。也可以通过牛顿迭代得到。

多项式exp

观察泰勒展开式：ef(x)=∑i=0nf(x)ii!e^{f(x)}=\sum_{i=0}^n\frac{f(x)^i}{i!}

定义f(i)f(i)为i个小球放入盒子的方案数，如果该盒子无序，那么其要除以i!。有些时候，计算集合中无顺序的方案数非常好算，但是有顺序的方案数非常难算，那么就可以通过exp的逆变换ln来计算。

多项式ln

ln，exp 的求法都可以通过构造函数并让其牛顿迭代得到。

模板部分

FFT模板

含convolution的模板

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using i64 = long long;
using Z = std::complex<long double>;
const double pi = std::acos(-1);
std::vector<int> rev;
std::vector<Z> roots{(0, 1), (0, 1)};

void dft(std::vector<Z>& a) {
	int n = a.size();

	if (int(rev.size()) != n) {
		rev.resize(n);
		for (int i = 0; i < n; ++i) {
			rev[i] = (rev[i >> 1] >> 1) | ((i & 1) ? n >> 1 : 0);
		}
	}

	for (int i = 0; i < n; ++i) {
		if (rev[i] < i)std::swap(a[i], a[rev[i]]);
	}
	if (int(roots.size() < n)) {
		int k = __builtin_ctz(roots.size());
		roots.resize(n);
		while ((1 << k) < n) {
			Z e(cos(acos(-1) / (1 << k)), sin(acos(-1) / (1 << k)));
			for (int i = 1 << (k - 1); i < (1 << k); i++) {
				roots[i << 1] = roots[i];
				roots[i << 1 | 1] = roots[i] * e;
			}
			k++;
		}
	}

	for (int k = 1; k < n; k *= 2) {
		for (int i = 0; i < n; i += 2 * k) {
			for (int j = 0; j < k; j++) {
				Z u = a[i + j];
				Z v = a[i + j + k] * roots[k + j];
				a[i + j] = u + v;
				a[i + j + k] = u - v;
			}
		}
	}
}

void idft(std::vector<Z>& a) {
	int n = a.size();
	std::reverse(a.begin() + 1, a.end());
	dft(a);
}

struct Poly : public std::vector<i64> {
	using std::vector<i64>::vector;

	Poly() {}

	Poly(const std::vector<i64>& a) : std::vector<i64>(a) {}

	Poly(const std::initializer_list<i64>& a) : std::vector<i64>(a) {}

	i64 operator [](int idx) const {
		if (idx > size())return 0;
		else return *(begin() + idx);
	}

	i64& operator [](int idx) { return *(begin() + idx); }

	friend Poly operator +(const Poly& a, const Poly& b) {
		Poly res(std::max(a.size(), b.size()));
		for (int i = 0; i < int(res.size()); i++) {
			res[i] = a[i] + b[i];
		}
		return res;
	}

	friend Poly operator -(const Poly& a, const Poly& b) {
		Poly res(std::max(a.size(), b.size()));
		for (int i = 0; i < int(res.size()); i++) {
			res[i] = a[i] - b[i];
		}
		return res;
	}

	friend Poly operator *(const Poly& a, const Poly& b) {
		if (a.size() == 0 || b.size() == 0) {
			return Poly();
		}
		int sz = 1, tot = a.size() + b.size() - 1;
		while (sz < tot) {
			sz *= 2;
		}
		std::vector<Z> f(sz);
		for (int i = 0; i < a.size(); ++i)f[i].real(a[i]);
		for (int i = 0; i < b.size(); ++i)f[i].imag(b[i]);
		dft(f);
		for (int i = 0; i < sz; ++i) {
			f[i] = f[i] * f[i];
			f[i] /= sz, f[i] /= 2;
		}
		idft(f);
		Poly ans(tot);
		for (int i = 0; i < tot; ++i)ans[i] = f[i].imag() + 0.5;
		return ans;
	}

	friend Poly operator *(i64 a, Poly b) {
		for (int i = 0; i < int(b.size()); i++) {
			b[i] *= a;
		}
		return b;
	}

	friend Poly operator *(Poly a, i64 b) {
		for (int i = 0; i < int(a.size()); i++) {
			a[i] *= b;
		}
		return a;
	}

	Poly operator <<(const int k) const {
		auto b = *this;
		b.insert(b.begin(), k, 0);
		return b;
	}

	Poly operator >>(const int k) const {
		if (size() <= k) {
			return {};
		}
		return {begin() + k, end()};
	}

	Poly& operator >>=(const int k) { return (*this) = (*this) >> k; }

	Poly& operator <<=(const int k) { return (*this) = (*this) << k; }

	Poly modxk(int k) const {
		k = std::min(k, (int) size());
		return {begin(), begin() + k};
	}

	Poly& operator +=(const Poly& b) { return (*this) = (*this) + b; }

	Poly& operator -=(const Poly& b) { return (*this) = (*this) - b; }

	Poly& operator *=(const Poly& b) { return (*this) = (*this) * b; }
};

NTT模板

含有ln,exp,inv,integal,derivation,convolution的模板

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template<typename T>
T inverse(T a, T b) {
    T u = 0, v = 1;
    while (a != 0) {
        T t = b / a;
        b -= t * a;
        std::swap(a, b);
        u -= t * v;
        std::swap(u, v);
    }
    assert(b == 1);
    return u;
}

template<typename T>
T power(T a, int b) {
    T ans = 1;
    for (; b; a *= a, b >>= 1) {
        if (b & 1)ans *= a;
    }
    return ans;
}

template<int Mod>
class Modular {
public:
    using Type = int;

    template<typename U>
    static Type norm(U &x) {
        Type v;
        if (-Mod <= x && x < Mod) v = static_cast<Type>(x);
        else v = static_cast<Type>(x % Mod);
        if (v < 0) v += Mod;
        return v;
    }

    constexpr Modular() : value() {}

    int val() const { return value; }

    Modular inv() const {
        return Modular(inverse(value, Mod));
    }

    template<typename U>
    Modular(const U &x) {
        value = norm(x);
    }

    const Type &operator()() const {
        return value;
    }

    template<typename U>
    explicit operator U() const {
        return static_cast<U>(value);
    }

    Modular &operator+=(const Modular &other) {
        if ((value += other.value) >= Mod) value -= Mod;
        return *this;
    }

    Modular &operator-=(
            const Modular &other) {
        if ((value -= other.value) < 0) value += Mod;
        return *this;
    }

    template<typename U>
    Modular &operator+=(const U &other) { return *this += Modular(other); }

    template<typename U>
    Modular &operator-=(const U &other) { return *this -= Modular(other); }

    Modular &operator++() { return *this += 1; }

    Modular &operator--() { return *this -= 1; }

    Modular operator++(int) {
        Modular result(*this);
        *this += 1;
        return result;
    }

    Modular operator--(int) {
        Modular result(*this);
        *this -= 1;
        return result;
    }

    Modular operator-() const { return Modular(-value); }

    template<class ISTREAM_TYPE>
    friend ISTREAM_TYPE &operator>>(ISTREAM_TYPE &is, Modular &rhs) {
        ll v;
        is >> v;
        rhs = Modular(v);
        return is;
    }

    template<class OSTREAM_TYPE>
    friend OSTREAM_TYPE &operator<<(OSTREAM_TYPE &os, const Modular &rhs) {
        return os << rhs.val();
    }

    Modular &operator*=(const Modular &rhs) {
        value = ll(value) * rhs.value % Mod;
        return *this;
    }

    Modular &operator/=(const Modular &other) { return *this *= Modular(inverse(other.value, Mod)); }

    friend const Type &abs(const Modular &x) { return x.value; }

    friend bool operator==(const Modular &lhs, const Modular &rhs) { return lhs.x == rhs.x; }

    friend bool operator<(const Modular &lhs, const Modular &rhs) { return lhs.x < rhs.x; }


    bool operator==(const Modular &rhs) { return *this == rhs.value; }

    template<typename U>
    bool operator==(U rhs) { return *this == Modular(rhs); }

    template<typename U>
    friend bool operator==(U lhs, const Modular &rhs) { return Modular(lhs) == rhs; }

    bool operator!=(const Modular &rhs) { return *this != rhs; }

    template<typename U>
    bool operator!=(U rhs) { return *this != rhs; }

    template<typename U>
    friend bool operator!=(U lhs, const Modular &rhs) { return lhs != rhs; }

    bool operator<(const Modular &rhs) { return this->value < rhs.value; }

    Modular operator+(const Modular &rhs) { return Modular(*this) += rhs; }

    template<typename U>
    Modular operator+(U rhs) { return Modular(*this) += rhs; }

    template<typename U>
    friend Modular operator+(U lhs, const Modular &rhs) { return Modular(lhs) += rhs; }

    Modular operator-(const Modular &rhs) { return Modular(*this) -= rhs; }

    template<typename U>
    Modular operator-(U rhs) { return Modular(*this) -= rhs; }

    template<typename U>
    friend Modular operator-(U lhs, const Modular &rhs) { return Modular(lhs) -= rhs; }

    Modular operator*(const Modular &rhs) { return Modular(*this) *= rhs; }

    template<typename U>
    Modular operator*(U rhs) { return Modular(*this) *= rhs; }

    template<typename U>
    friend Modular operator*(U lhs, const Modular &rhs) { return Modular(lhs) *= rhs; }

    Modular operator/(const Modular &rhs) { return Modular(*this) /= rhs; }

    template<typename U>
    Modular operator/(U rhs) { return Modular(*this) /= rhs; }

    template<typename U>
    friend Modular operator/(U lhs, const Modular &rhs) { return Modular(lhs) /= rhs; }

private:
    Type value;
};

constexpr int mod = 998244353;
using Z = Modular<mod>;


std::vector<int> rev;
std::vector<Z> roots{0, 1};

void dft(std::vector<Z> &a) {
    int n = a.size();

    if (int(rev.size()) != n) {
        rev.resize(n);
        for (int i = 0; i < n; ++i) {
            rev[i] = (rev[i >> 1] >> 1) | ((i & 1) ? n >> 1 : 0);
        }
    }

    for (int i = 0; i < n; ++i) {
        if (rev[i] < i)std::swap(a[i], a[rev[i]]);
    }
    if (int(roots.size() < n)) {
        int k = __builtin_ctz(roots.size());
        roots.resize(n);
        while ((1 << k) < n) {
            Z e = power(Z(3), (mod - 1) >> (k + 1));
            for (int i = 1 << (k - 1); i < (1 << k); i++) {
                roots[i << 1] = roots[i];
                roots[i << 1 | 1] = roots[i] * e;
            }
            k++;
        }
    }

    for (int k = 1; k < n; k *= 2) {
        for (int i = 0; i < n; i += 2 * k) {
            for (int j = 0; j < k; j++) {
                Z u = a[i + j];
                Z v = a[i + j + k] * roots[k + j];
                a[i + j] = u + v;
                a[i + j + k] = u - v;
            }
        }
    }
}

void idft(std::vector<Z> &a) {
    int n = a.size();
    std::reverse(a.begin() + 1, a.end());
    dft(a);
}

struct Poly : public std::vector<Z> {
    using std::vector<Z>::vector;

    Poly() = default;

    explicit Poly(const std::vector<Z> &a) : vector(a) {}

    friend Poly operator+(const Poly &a, const Poly &b) {
        Poly res(std::max(a.size(), b.size()));
        for (int i = 0; i < int(res.size()); i++) {
            if (i < a.size())res[i] = a[i];
            if (i < b.size())res[i] += b[i];
        }
        return res;
    }

    friend Poly operator-(const Poly &a, const Poly &b) {
        Poly res(std::max(a.size(), b.size()));
        for (int i = 0; i < int(res.size()); i++) {
            if (i < a.size()) res[i] = a[i];
            if (i < b.size()) res[i] -= b[i];
        }

        return res;
    }

    friend Poly operator*(const Poly &a, const Poly &b) {
        if (a.empty() || b.empty()) {
            return {};
        }
        int sz = 1, tot = a.size() + b.size() - 1;
        while (sz < tot) {
            sz *= 2;
        }
        std::vector<Z> f(a), g(b);
        f.resize(sz), g.resize(sz);
        dft(f), dft(g);
        for (int i = 0; i < sz; ++i) {
            f[i] = f[i] * g[i];
            f[i] /= sz;
        }
        idft(f);
        return Poly(f).modxk(tot);
    }

    friend Poly operator*(int a, Poly b) {
        for (int i = 0; i < int(b.size()); i++) {
            b[i] *= a;
        }
        return b;
    }

    friend Poly operator*(Poly a, int b) {
        for (int i = 0; i < int(a.size()); i++) {
            a[i] *= b;
        }
        return a;
    }

    Poly operator<<(const int k) const {
        auto b = *this;
        b.insert(b.begin(), k, 0);
        return b;
    }

    Poly operator>>(const int k) const {
        if (size() <= k) {
            return {};
        }
        return Poly(begin() + k, end());
    }

    Poly &operator>>=(const int k) {
        return (*this) = (*this) >> k;
    }

    Poly &operator<<=(const int k) {
        return (*this) = (*this) << k;
    }

    Poly mulxk(int k) const {
        return *this << k;
    }

    Poly modxk(int k) const {
        k = std::min(k, (int) size());
        return Poly(begin(), begin() + k);
    }

    Poly divxk(int k) const {
        if (size() <= k) {
            return {};
        }
        return Poly(begin() + k, end());
    }

    Poly &operator+=(const Poly &b) {
        return (*this) = (*this) + b;
    }

    Poly &operator-=(const Poly &b) {
        return (*this) = (*this) - b;
    }

    Poly &operator*=(const Poly &b) {
        return (*this) = (*this) * b;
    }


    Poly deriv() const {
        if (empty()) {
            return {};
        }
        Poly res(size() - 1);
        for (int i = 0; i < size() - 1; ++i) {
            res[i] = (i + 1) * (*this)[i + 1];
        }
        return res;
    }

    Poly integr() const {
        Poly res(size() + 1);
        for (int i = 0; i < size(); ++i) {
            res[i + 1] = (*this)[i] / (i + 1);
        }
        return res;
    }

    Poly inv(int m) const {
        Poly x{(*this)[0].inv()};
        int k = 1;
        while (k < m) {
            k *= 2;
            x = (x * (Poly{2} - modxk(k) * x)).modxk(k);
        }
        return x.modxk(m);
    }

    Poly log(int m) const {
        return (deriv() * inv(m)).integr().modxk(m);
    }

    Poly exp(int m) const {
        Poly x{Z(1)};
        int k = 1;
        while (k < m) {
            k *= 2;
            x = (x * (Poly{Z(1)} - x.log(k) + modxk(k))).modxk(k);
        }
        return x.modxk(m);
    }

    Poly pow(int k, int m) const {
        int i = 0;
        while (i < size() && (*this)[i].val() == 0) {
            i++;
        }
        if (i == size() || 1LL * i * k >= m) {
            return Poly(m);
        }
        Z v = (*this)[i];
        Poly f = divxk(i) * (v.inv().val());
        return (f.log(m - i * k) * k).exp(m - i * k).mulxk(i * k) * power(v, k).val();
    }

    Poly sqrt(int m) const {
        Poly x{Z(1)};
        int k = 1;
        while (k < m) {
            k *= 2;
            x = (x + (modxk(k) * x.inv(k)).modxk(k)) * ((mod + 1) / 2);
        }
        return x.modxk(m);
    }

    Poly mulT(Poly b) const {
        if (b.empty()) {
            return {};
        }
        int n = b.size();
        std::reverse(b.begin(), b.end());
        return ((*this) * b).divxk(n - 1);
    }

    std::vector<Z> eval(std::vector<Z> x) const {
        if (empty()) {
            return Poly(x.size());
        }
        const int n = std::max(int(x.size()), (int) size());
        std::vector<Poly> q(4 * n);
        std::vector<Z> ans(x.size());
        x.resize(n);
        std::function<void(int, int, int)> build = [&](int p, int l, int r) {
            if (r - l == 1) {
                q[p] = Poly{1, -x[l]};
            } else {
                int m = (l + r) / 2;
                build(2 * p, l, m);
                build(2 * p + 1, m, r);
                q[p] = q[2 * p] * q[2 * p + 1];
            }
        };
        build(1, 0, n);
        std::function<void(int, int, int, const Poly &)> work = [&](int p, int l, int r, const Poly &num) {
            if (r - l == 1) {
                if (l < int(ans.size())) {
                    ans[l] = num[0];
                }
            } else {
                int m = (l + r) / 2;
                work(2 * p, l, m, num.mulT(q[2 * p + 1]).modxk(m - l));
                work(2 * p + 1, m, r, num.mulT(q[2 * p]).modxk(r - m));
            }
        };
        work(1, 0, n, mulT(q[1].inv(n)));
        return ans;
    }
};

好抄一点的板子:

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using i64 = long long;
constexpr int P = 998244353;


template<class T>
T inverse(T a, T b) {
	T u = 0, v = 1;
	while (a != 0) {
		T t = b / a;
		b -= t * a;
		std::swap(a, b);
		u -= t * v;
		std::swap(u, v);
	}
	assert(b == 1);
	return u;
}

template<class T>
T power(T a, i64 b) {
	T ret = 1;
	for (; b; b >>= 1, a = a * a)if (b & 1)ret = ret * a;
	return ret;
}

int norm(int x) {
	return x < 0 ? x += P : x >= P ? x -= P : x;
}

struct ModInt {
	int a;

	ModInt(int a = 0) : a(norm(a)) {} // assume a <= 2P
	ModInt(i64 a) : a(norm(a % P)) {}

	int val() const { return a; }

	ModInt inv() const { return power(*this, P - 2); }

	ModInt operator -() const { return ModInt(norm(P - a)); }

	friend ModInt operator *(const ModInt& lhs, const ModInt& rhs) { return {i64(lhs.a) * rhs.a}; }

	friend ModInt operator +(const ModInt& lhs, const ModInt& rhs) { return {lhs.a + rhs.a}; }

	friend ModInt operator -(const ModInt& lhs, const ModInt& rhs) { return {lhs.a - rhs.a}; }

	friend ModInt operator /(const ModInt& lhs, const ModInt& rhs) { return {i64(lhs.a) * rhs.inv()}; }

	friend std::istream& operator >>(std::istream& is, ModInt& rhs) {
		i64 x;
		is >> x;
		rhs = ModInt(x);
		return is;
	}

	friend std::ostream& operator <<(std::ostream& os, const ModInt& rhs) { return os << rhs.val(); }
};

using Z = ModInt;
std::vector<int> rev;
std::vector<Z> roots{0, 1};

void dft(std::vector<Z>& a) {
	int n = a.size();

	if (int(rev.size()) != n) {
		rev.resize(n);
		for (int i = 0; i < n; ++i) {
			rev[i] = (rev[i >> 1] >> 1) | ((i & 1) ? n >> 1 : 0);
		}
	}

	for (int i = 0; i < n; ++i) {
		if (rev[i] < i)std::swap(a[i], a[rev[i]]);
	}
	if (int(roots.size() < n)) {
		int k = __builtin_ctz(roots.size());
		roots.resize(n);
		while ((1 << k) < n) {
			Z e = power(Z(3), (P - 1) >> (k + 1));
			for (int i = 1 << (k - 1); i < (1 << k); i++) {
				roots[i << 1] = roots[i];
				roots[i << 1 | 1] = roots[i] * e;
			}
			k++;
		}
	}

	for (int k = 1; k < n; k *= 2) {
		for (int i = 0; i < n; i += 2 * k) {
			for (int j = 0; j < k; j++) {
				Z u = a[i + j];
				Z v = a[i + j + k] * roots[k + j];
				a[i + j] = u + v;
				a[i + j + k] = u - v;
			}
		}
	}
}

void idft(std::vector<Z>& a) {
	int n = a.size();
	std::reverse(a.begin() + 1, a.end());
	dft(a);
}

struct Poly : public std::vector<Z> {
	using std::vector<Z>::vector;

	Poly() = default;

	explicit Poly(const std::vector<Z>& a) : vector(a) {}

	Z operator [](int idx) const {
		if (idx < size()) {
			return *(this->begin() + idx);
		} else {
			return 0;
		}
	}

	Z& operator [](int idx) {
		return *(this->begin() + idx);
	}

	friend Poly operator +(const Poly& a, const Poly& b) {
		Poly res(std::max(a.size(), b.size()));
		for (int i = 0; i < int(res.size()); i++) {
			res[i] = a[i] + b[i];
		}
		return res;
	}

	friend Poly operator -(const Poly& a, const Poly& b) {
		Poly res(std::max(a.size(), b.size()));
		for (int i = 0; i < int(res.size()); i++) {
			res[i] = a[i] - b[i];
		}

		return res;
	}

	friend Poly operator *(Poly a, Poly b) {
		if (a.empty() || b.empty()) {
			return {};
		}
		int sz = 1, tot = a.size() + b.size() - 1;
		while (sz < tot) {
			sz *= 2;
		}
		a.resize(sz), b.resize(sz);
		dft(a), dft(b);
		Z inv = (1 - P) / sz;
		for (int i = 0; i < sz; ++i) {
			a[i] = a[i] * b[i];
			a[i] = a[i] * inv;
		}
		idft(a);
		return Poly(a).modxk(tot);
	}

	friend Poly operator *(int a, Poly b) {
		for (int i = 0; i < int(b.size()); i++) {
			b[i] = b[i] * a;
		}
		return b;
	}

	friend Poly operator *(Poly a, int b) {
		for (int i = 0; i < int(a.size()); i++) {
			a[i] = a[i] * b;
		}
		return a;
	}

	Poly operator <<(const int k) const {
		auto b = *this;
		b.insert(b.begin(), k, 0);
		return b;
	}

	Poly operator >>(const int k) const {
		if (size() <= k) {
			return {};
		}
		return Poly(begin() + k, end());
	}

	Poly mulxk(int k) const {
		return *this << k;
	}

	Poly modxk(int k) const {
		k = std::min(k, (int) size());
		return Poly(begin(), begin() + k);
	}

	Poly divxk(int k) const {
		if (size() <= k) {
			return {};
		}
		return Poly(begin() + k, end());
	}


	Poly deriv() const {
		if (empty()) {
			return {};
		}
		Poly res(size() - 1);
		for (int i = 0; i < size() - 1; ++i) {
			res[i] = (i + 1) * (*this)[i + 1];
		}
		return res;
	}

	Poly integr() const {
		Poly res(size() + 1);
		for (int i = 0; i < size(); ++i) {
			res[i + 1] = (*this)[i] / (i + 1);
		}
		return res;
	}

	Poly inv(int m) const {
		Poly x{(*this)[0].inv()};
		int k = 1;
		while (k < m) {
			k *= 2;
			x = (x * (Poly{2} - modxk(k) * x)).modxk(k);
		}
		return x.modxk(m);
	}

	Poly log(int m) const {
		return (deriv() * inv(m)).integr().modxk(m);
	}

	Poly exp(int m) const {
		Poly x{Z(1)};
		int k = 1;
		while (k < m) {
			k *= 2;
			x = (x * (Poly{Z(1)} - x.log(k) + modxk(k))).modxk(k);
		}
		return x.modxk(m);
	}

	Poly pow(int k, int m) const {
		int i = 0;
		while (i < size() && (*this)[i].val() == 0) {
			i++;
		}
		if (i == size() || 1LL * i * k >= m) {
			return Poly(m);
		}
		Z v = (*this)[i];
		Poly f = divxk(i) * (v.inv().val());
		return (f.log(m - i * k) * k).exp(m - i * k).mulxk(i * k) * power(v, k).val();
	}

	Poly sqrt(int m) const {
		Poly x{Z(1)};
		int k = 1;
		while (k < m) {
			k *= 2;
			x = (x + (modxk(k) * x.inv(k)).modxk(k)) * ((P + 1) / 2);
		}
		return x.modxk(m);
	}

	Poly mulT(Poly b) const {
		if (b.empty()) {
			return {};
		}
		int n = b.size();
		std::reverse(b.begin(), b.end());
		return ((*this) * b).divxk(n - 1);
	}

	std::vector<Z> eval(std::vector<Z> x) const {
		if (empty()) {
			return Poly(x.size());
		}
		const int n = std::max(int(x.size()), (int) size());
		std::vector<Poly> q(4 * n);
		std::vector<Z> ans(x.size());
		x.resize(n);
		std::function<void(int, int, int)> build = [&](int p, int l, int r) {
			if (r - l == 1) {
				q[p] = Poly{1, -x[l]};
			} else {
				int m = (l + r) / 2;
				build(2 * p, l, m);
				build(2 * p + 1, m, r);
				q[p] = q[2 * p] * q[2 * p + 1];
			}
		};
		build(1, 0, n);
		std::function<void(int, int, int, const Poly&)> work = [&](int p, int l, int r, const Poly& num) {
			if (r - l == 1) {
				if (l < int(ans.size())) {
					ans[l] = num[0];
				}
			} else {
				int m = (l + r) / 2;
				work(2 * p, l, m, num.mulT(q[2 * p + 1]).modxk(m - l));
				work(2 * p + 1, m, r, num.mulT(q[2 * p]).modxk(r - m));
			}
		};
		work(1, 0, n, mulT(q[1].inv(n)));
		return ans;
	}
};

MTT模板

含 convolution,inv 拆系数fft实现的板子

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template<typename T>
T inverse(T a, T b) {
	T u = 0, v = 1;
	while (a != 0) {
		T t = b / a;
		b -= t * a;
		std::swap(a, b);
		u -= t * v;
		std::swap(u, v);
	}
	assert(b == 1);
	return u;
}

template<typename T>
T power(T a, long long b, int mod) {
	T ans = 1;
	for (; b; a = 1ll * a * a % mod, b >>= 1) {
		if (b & 1)ans = 1ll * ans * a % mod;
	}
	return ans;
}

int mod;
using Z = std::complex<long double>;
using i64 = long long;
const double pi = std::acos(-1);
std::vector<int> rev;
std::vector<Z> roots{(0, 1), (0, 1)};

int getsz(int x) {
	int sz = 1;
	while (sz < x)sz *= 2;
	return sz;
}

void dft(std::vector<Z>& a) {
	int n = a.size();

	if (int(rev.size()) != n) {
		rev.resize(n);
		for (int i = 0; i < n; ++i) {
			rev[i] = (rev[i >> 1] >> 1) | ((i & 1) ? n >> 1 : 0);
		}
	}

	for (int i = 0; i < n; ++i) {
		if (rev[i] < i)std::swap(a[i], a[rev[i]]);
	}
	if (int(roots.size() < n)) {
		int k = __builtin_ctz(roots.size());
		roots.resize(n);
		while ((1 << k) < n) {
			Z e(cos(acos(-1) / (1 << k)), sin(acos(-1) / (1 << k)));
			for (int i = 1 << (k - 1); i < (1 << k); i++) {
				roots[i << 1] = roots[i];
				roots[i << 1 | 1] = roots[i] * e;
			}
			k++;
		}
	}

	for (int k = 1; k < n; k *= 2) {
		for (int i = 0; i < n; i += 2 * k) {
			for (int j = 0; j < k; j++) {
				Z u = a[i + j];
				Z v = a[i + j + k] * roots[k + j];
				a[i + j] = u + v;
				a[i + j + k] = u - v;
			}
		}
	}
}

void idft(std::vector<Z>& a) {
	int n = a.size();
	std::reverse(a.begin() + 1, a.end());
	dft(a);
}

template<typename U>
static U norm(U& x, int Mod) {
	if (-Mod <= x && x < Mod) x = static_cast<U>(x);
	else x = static_cast<U>(x % Mod);
	if (x < 0) x += Mod;
	return x;
}

struct Poly : public std::vector<i64> {
	using std::vector<i64>::vector;

	i64 operator [](int idx) const {
		if (idx > size())return 0;
		else return *(begin() + idx);
	}

	i64& operator [](int idx) { return *(begin() + idx); }

	friend Poly operator +(const Poly& a, const Poly& b) {
		Poly res(std::max(a.size(), b.size()));
		for (int i = 0; i < int(res.size()); i++) {
			res[i] = a[i] + b[i];
			norm(res[i], mod);
		}
		return res;
	}

	friend Poly operator -(const Poly& a, const Poly& b) {
		Poly res(std::max(a.size(), b.size()));
		for (int i = 0; i < int(res.size()); i++) {
			res[i] = a[i] - b[i];
			norm(res[i], mod);
		}
		return res;
	}

	friend Poly operator *(int a, Poly b) {
		for (int i = 0; i < int(b.size()); i++) {
			b[i] *= a;
		}
		return b;
	}

	friend Poly operator *(Poly a, int b) {
		for (int i = 0; i < int(a.size()); i++) {
			a[i] *= b;
		}
		return a;
	}

	Poly operator <<(const int k) const {
		auto b = *this;
		b.insert(b.begin(), k, 0);
		return b;
	}

	Poly operator >>(const int k) const {
		if (size() <= k) {
			return {};
		}
		return Poly(begin() + k, end());
	}

	Poly& operator >>=(const int k) {
		return (*this) = (*this) >> k;
	}

	Poly& operator <<=(const int k) {
		return (*this) = (*this) << k;
	}

	Poly mulxk(int k) const {
		return *this << k;
	}

	Poly modxk(int k) const {
		k = std::min(k, (int) size());
		return Poly(begin(), begin() + k);
	}

	Poly divxk(int k) const {
		if (size() <= k) {
			return {};
		}
		return Poly(begin() + k, end());
	}

	Poly& operator +=(const Poly& b) {
		return (*this) = (*this) + b;
	}

	Poly& operator -=(const Poly& b) {
		return (*this) = (*this) - b;
	}

	Poly& operator *=(const Poly& b) {
		return (*this) = (*this) * b;
	}


	Poly deriv() const {
		if (empty()) {
			return {};
		}
		Poly res(size() - 1);
		for (int i = 0; i < size() - 1; ++i) {
			res[i] = (i + 1) * (*this)[i + 1] % mod;
		}
		return res;
	}

	Poly integr() const {
		Poly res(size() + 1);
		for (int i = 0; i < size(); ++i) {
			res[i + 1] = (*this)[i] * power(i + 1, mod - 2, mod);
		}
		return res;
	}

	friend Poly operator *(const Poly& x, const Poly& y) {
		if (x.empty() || y.empty())return {};
		Poly a(x), b(y);
		int len = a.size() + b.size() - 1, n = getsz(len);
		vector<Z> f(n), g(n), p(n), q(n);
		for (int i = 0; i < a.size(); i++)
			f[i] = Z(a[i] >> 15, a[i] & 0x7fff);
		for (int i = 0; i < b.size(); i++)
			g[i] = Z(b[i] >> 15, b[i] & 0x7fff);
		dft(f), dft(g);
		for (int i = 0; i < n; ++i) {
			int r = (n - i) & (n - 1);
			p[i] = Z(0.5 * (f[i].real() + f[r].real()), 0.5 * (f[i].imag() - f[r].imag())) * g[i];
			q[i] = Z(0.5 * (f[i].imag() + f[r].imag()), 0.5 * (f[r].real() - f[i].real())) * g[i];
		}
		idft(p), idft(q);
		for (int i = 0; i < n; ++i)p[i] /= n, q[i] /= n;
		a.resize(len);
		for (int i = 0; i < len; i++) {
			long long X, Y, Z, W;
			X = p[i].real() + 0.5, Y = p[i].imag() + 0.5;
			Z = q[i].real() + 0.5, W = q[i].imag() + 0.5;
			a[i] = ((X % mod << 30) + ((Y + Z) % mod << 15) + W) % mod;
		}
		return a;
	}

	Poly inv(int m) const {
		Poly x{(power((*this)[0], mod - 2, mod))};
		int k = 1;
		while (k < m) {
			k <<= 1;
			x = (x * (Poly{2} - modxk(k) * x)).modxk(k);
		}
		return x;
	}
};

acm > 数学 > 多项式 > 模板
#数学 #多项式
多项式板子
https://mrxyan6.github.io/2022/09/04/poly-templates/
作者
mrx
发布于
2022年9月4日
许可协议